Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is a polynomial equation of degree 2. In the IGCSE Edexcel syllabus, you must be able to solve quadratic equations using factorisation, the quadratic formula, and completing the square. You should also understand the discriminant and how the roots relate to the graph of the function.

二次方程是次数为2的多项式方程。在IGCSE Edexcel考纲中，你必须掌握用因式分解、求根公式和配方法解二次方程，同时理解判别式以及根与函数图像之间的关系。

1. The Standard Form | 标准形式

A quadratic equation is usually written in the standard form:

二次方程通常写成标准形式：

ax² + bx + c = 0, 其中 a ≠ 0

Here, a, b, and c are constants, and x is the unknown variable. The value of a must not be zero, otherwise the equation becomes linear.

这里 a、b、c 是常数，x 是未知数。a 不能为零，否则方程就变成一次方程了。

For example, 2x² – 5x + 3 = 0 is a quadratic equation with a = 2, b = -5, c = 3.

例如，2x² – 5x + 3 = 0 是一个二次方程，其中 a = 2，b = -5，c = 3。

2. Solving by Factorisation | 因式分解法

If the quadratic expression can be factorised, we can write it as a product of two linear factors. Then we set each factor equal to zero.

如果二次表达式能够因式分解，我们可以把它写成两个一次因式的乘积，然后令每个因式等于零。

Step 1: Write the equation in the form ax² + bx + c = 0.

第一步：把方程写成 ax² + bx + c = 0 的形式。

Step 2: Factorise the left-hand side.

第二步：对左边进行因式分解。

Step 3: Apply the zero product property: if pq = 0, then p = 0 or q = 0.

第三步：利用零乘积性质：若 pq = 0，则 p = 0 或 q = 0。

Example: Solve x² – 7x + 10 = 0.

例：解 x² – 7x + 10 = 0。

Factorise: (x – 2)(x – 5) = 0.

因式分解：(x – 2)(x – 5) = 0。

Therefore x – 2 = 0 or x – 5 = 0, giving x = 2 or x = 5.

因此 x – 2 = 0 或 x – 5 = 0，得到 x = 2 或 x = 5。

Always check by substituting the roots back into the original equation.

一定要把根代回原方程检验。

3. The Quadratic Formula | 求根公式

When the quadratic cannot be factorised easily, use the quadratic formula. For ax² + bx + c = 0:

当二次式不易因式分解时，使用求根公式。对于 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / (2a)

This formula gives the roots directly. It works for all quadratic equations, provided the discriminant is not negative.

该公式直接给出根。只要判别式不为负，它适用于所有二次方程。

Example: Solve 2x² + 3x – 2 = 0 using the formula.

例：用公式解 2x² + 3x – 2 = 0。

Here a = 2, b = 3, c = -2.

这里 a = 2，b = 3，c = -2。

Substitute: x = (−3 ± √(3² − 4×2×(−2))) / (2×2) = (−3 ± √(9 + 16)) / 4 = (−3 ± √25) / 4.

代入：x = (−3 ± √(3² − 4×2×(−2))) / (2×2) = (−3 ± √(9 + 16)) / 4 = (−3 ± √25) / 4。

So x = (−3 + 5)/4 = 0.5 or x = (−3 − 5)/4 = −2.

所以 x = (−3 + 5)/4 = 0.5 或 x = (−3 − 5)/4 = −2。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This is useful for solving equations and for finding turning points.

配方法把二次表达式改写为 a(x + p)² + q 的形式。它常用于解方程和求顶点。

For the equation x² + 6x – 7 = 0, complete the square:

对于方程 x² + 6x – 7 = 0，配方：

Half of 6 is 3, so (x + 3)² = x² + 6x + 9.

6 的一半是 3，所以 (x + 3)² = x² + 6x + 9。

Therefore x² + 6x – 7 = (x + 3)² – 9 – 7 = (x + 3)² – 16.

因此 x² + 6x – 7 = (x + 3)² – 9 – 7 = (x + 3)² – 16。

Set the expression equal to zero: (x + 3)² – 16 = 0.

令表达式为零：(x + 3)² – 16 = 0。

Then (x + 3)² = 16, so x + 3 = ±4, giving x = 1 or x = -7.

于是 (x + 3)² = 16，所以 x + 3 = ±4，得到 x = 1 或 x = -7。

For a general quadratic ax² + bx + c, first factor out a from the x terms, then complete the square inside the bracket.

对于一般二次方程 ax² + bx + c，先从含 x 的项中提出 a，再在括号内配方。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: Δ = b² – 4ac.

判别式是求根公式中平方根下的部分：Δ = b² – 4ac。

It tells us how many real roots the equation has.

它告诉我们方程有多少个实数根。

判别式 Δ	实根个数
Δ > 0	两个不同的实根
Δ = 0	两个相等的实根（一个重根）
Δ < 0	没有实数根

If Δ ≥ 0, the roots are real. If Δ > 0, the graph crosses the x-axis twice; if Δ = 0, it touches the x-axis once; if Δ < 0, it never touches the x-axis.

若 Δ ≥ 0，根为实数。若 Δ > 0，图像与 x 轴有两个交点；若 Δ = 0，图像与 x 轴相切；若 Δ < 0，图像与 x 轴无交点。

6. Sum and Product of Roots | 根与系数的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β, we have:

对于二次方程 ax² + bx + c = 0，若其根为 α 和 β，则有：

α + β = −b/a

αβ = c/a

These relationships are derived from expanding (x − α)(x − β) = 0.

这些关系由展开 (x − α)(x − β) = 0 得到。

Example: Find the sum and product of the roots of 3x² – 6x + 2 = 0.

例：求 3x² – 6x + 2 = 0 的根之和与根之积。

The sum is −b/a = −(−6)/3 = 2, and the product is c/a = 2/3.

根之和为 −b/a = −(−6)/3 = 2，根之积为 c/a = 2/3。

7. Constructing Equations from Roots | 由根构造方程

If you know the roots α and β, you can write the quadratic equation as:

若已知根 α 和 β，可以写出二次方程：

x² − (α + β)x + αβ = 0

This is called the monic quadratic with given roots. If a leading coefficient is required, multiply the whole equation by any constant k.

这就是给定根的首一二次方程。如果需要首项系数，可将整个方程乘以任意常数 k。

Example: Construct a quadratic equation with roots 3 and -4.

例：构造一个以 3 和 -4 为根的二次方程。

Sum = 3 + (−4) = −1. Product = 3 × (−4) = −12.

和 = 3 + (−4) = −1。积 = 3 × (−4) = −12。

The equation is x² − (−1)x + (−12) = 0, i.e. x² + x − 12 = 0.

方程为 x² − (−1)x + (−12) = 0，即 x² + x − 12 = 0。

8. Graphical Interpretation | 图像解释

The graph of y = ax² + bx + c is a parabola. The x-intercepts of the graph are the real roots of the equation ax² + bx + c = 0.

函数 y = ax² + bx + c 的图像是一条抛物线。图像与 x 轴交点的横坐标就是方程 ax² + bx + c = 0 的实数根。

If a > 0, the parabola opens upward; if a < 0, it opens downward.

若 a > 0，抛物线开口向上；若 a < 0，抛物线开口向下。

The vertex (turning point) has x-coordinate x = −b/(2a). Substitute this into the equation to find the y-coordinate.

顶点（驻点）的 x 坐标为 x = −b/(2a)。将其代入方程可求得 y 坐标。

Example: For y = x² – 4x + 3, the x-intercepts are found by solving x² – 4x + 3 = 0, which factorises as (x – 1)(x – 3) = 0, so x = 1 and x = 3.

例：对于 y = x² – 4x + 3，通过解 x² – 4x + 3 = 0 求 x 轴交点，因式分解为 (x – 1)(x – 3) = 0，所以 x = 1 和 x = 3。

The vertex is at x = 4/(2×1) = 2, and y = 2² – 8 + 3 = −1, so the vertex is (2, −1).

顶点在 x = 4/(2×1) = 2，y = 2² – 8 + 3 = −1，所以顶点为 (2, −1)。

9. Word Problems | 应用题

Quadratic equations often appear in geometry and motion problems. Set up the equation from the given information, then solve it using the most suitable method.

二次方程常出现在几何和运动问题中。根据已知信息列出方程，然后选择最合适的方法求解。

Example: A rectangle has length 4 cm longer than its width. Its area is 96 cm². Find the width.

例：一个矩形的长比宽多 4 cm，面积为 96 cm²。求宽。

Let the width be x cm. Then the length is (x + 4) cm. Area = x(x + 4) = 96.

设宽为 x cm，则长为 (x + 4) cm。面积 = x(x + 4) = 96。

Expanding gives x² + 4x – 96 = 0. Factorise: (x + 12)(x – 8) = 0.

展开得 x² + 4x – 96 = 0。因式分解：(x + 12)(x – 8) = 0。

Therefore x = -12 or x = 8. Since width cannot be negative, the width is 8 cm.

因此 x = -12 或 x = 8。由于宽度不能为负，所以宽为 8 cm。

Always reject any solution that does not make sense in the context.

始终舍去在情境中没有意义的解。

10. Common Mistakes | 常见错误

Make sure you avoid these frequent errors in exams.

在考试中务必避免这些常见错误。

Forgetting to rearrange the equation into ax² + bx + c = 0 before factorising or applying the formula.

在因式分解或使用公式前忘记把方程整理成 ax² + bx + c = 0 的形式。
Dividing both sides by x when x could be zero, which loses a root. For example, x² = 3x should be solved as x(x – 3) = 0, not x = 3.

当 x 可能为零时两边除以 x，从而丢根。例如 x² = 3x 应化为 x(x – 3) = 0 求解，而不是直接得 x = 3。
Misusing the quadratic formula signs: ensure b² − 4ac is calculated carefully, especially when c is negative.

使用求根公式时符号出错：务必仔细计算 b² − 4ac，特别是当 c 为负数时。
Ignoring the ± sign in the quadratic formula or in the square root step of completing the square.

忽略求根公式中的 ± 号或配方开方步骤中的 ± 号。

Double-check your factorisation by expanding it back. If a factorised answer expands to a different expression, your roots will be wrong.

通过展开来检查你的因式分解。如果展开结果与原来不同，那么根一定是错的。

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Solving Quadratic Equations | 解二次方程